Information Technology Reference
In-Depth Information
Ta b l e 7 . 1
Description of the
swaption pricing algorithm
Set
v
0
=
1.
For
j
=
2
,...,n
,
Solve (
7.6
) to compute bond price
B(T
1
,T
j
,r)
=
v(T
j
)
.
Next
j
Compute swap value
V
PFS
(T
1
,r)
by (
7.7
)
Set
v
0
V
PFS
(T
1
,r)
.
Solve (
7.6
) to compute swaption price
=
The counterpart of a future contract on stock markets are swap agreements on
interest rates. A
payer interest rate swap
is a contract exchanging fixed and variable
payments at certain time instances in the future, called the
tenor structure
.Let
t<
T
1
<
<T
n
<T
, then the value
V
PFS
(t, r)
is given as
···
r
,
n
e
−
T
i
V
PFS
(t, r)
r(s)
d
s
(T
i
−
= E
T
i
−
1
)(L(T
i
−
1
,T
i
,r
T
i
−
1
)
−
K)
|
r
t
=
t
i
=
2
whereas the value of a receiver interest rate swap is given as
V
RIS
(t, r)
=
−
V
PFS
(t, r)
. Using the definition of a simply compounded forward interest rate
(Definition
7.2.2
), we can view a payer interest rate swap as a portfolio of forward
rate agreements and obtain the following representation for the value
V
PFS
(t, r)
r
n
e
−
T
i
V
PFS
(t, r)
r(s)
d
s
(T
i
−
= E
T
i
−
1
)(F (t, T
i
−
1
,T
i
,r
t
)
−
K)
|
r
t
=
t
i
=
2
n
=
−
−
(T
i
−
B(t,T
1
,r)
B(t,T
n
,r)
T
i
−
1
)KB(t, T
i
,r).
(7.7)
i
=
2
Another example of derivatives on interest rates are swaptions. A
European payer
swaption
gives its holder the right but not the obligation to enter a payer interest
rate swap at a future time, the swaption maturity. The value
V(t,r)
of the payer
swaption, for which the maturity coincides with the first reset date
T
1
is given as
e
−
T
1
r(s)
d
s
V(t,r)
= E
t
K)
r
.
n
e
−
T
i
T
1
r(s)
d
s
(T
i
−
×
T
1
)(F (T
1
,T
i
−
1
,T
i
,r
T
1
)
−
|
r
t
=
i
=
2
+
In contrast to a swap, it cannot be decomposed into a portfolio of simpler products.
However, we may proceed as in the preceding chapter: first, we compute the value
of the swap contract underlying the swaption which can be priced using the PDE
(
7.2
). Second, the swaption can then be priced as a compound option, where the
underlying is the zero coupon bond. This is described schematically in Table
7.1
.
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